chapter 1 financial statistics

7/30/2019 Chapter 1 Financial statistics

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Financial Statistics

Master in FinanceUniversidad Carlos III de Madrid

Esther Ruiz and Diego Friesoli

2011-2012


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Chapter 1. Introduction: Basic concepts

The use of quantitative analysis to make better investment decisions

Mark J.P. Anson

Defusco et al. (2004)

Objective: Introduce some statistical tools useful for analyzing financial timeseries.

Outline:

1. Why quantitative tools are important for financial professionals

2. Differences between cross-sectional and time series data

3. Covariance and strict stationarity

4. Correlations and independence: differences between differencemartingala, white noise, strict white noise and Gaussian white noise

5. Describing variables: Unconditional and conditional moments

6. Linear and non-linear models


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1.1 Why quantitative tools are important for financial

professionals

Financial economics is a highly empirical discipline. Despite itsempirical nature, like other social sciences, financial

economics is almost entirely nonexperimental. Therefore,the primary method of inference for the financialeconomist is model-based statistical inference: Financialeconometrics.

The main distinction between econometrics in other areasand financial econometrics is the central role thatuncertainty plays in both financial theory and its empiricalimplementation: The substance of every financial modelinvolves the impact of uncertainty on the behaviour ofinvestors and, ultimately, on the the market prices.

Campbell, Lo and McKinlay (1997)


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Financial markets are very complicated places. There are many

interwoven variables that can affect the price of securities inan investment portfolio:

Macroeconomic factors: level of interest rates, currentaccount deficits, government spending and economic cycles.

Factors peculiar to the company: cash flow, working capital,book-to-market value, earning growth rates, dividend policy,debt-to-equity ratios.

Financial market variables: beta (measure of systematic risk).

, ,to earnings announcements, momentum trading.

Only quantitative techniques can help to understand the largenumber of plausible variables that can impact the price of a

security.


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1.2 Differences between cross-sectional and time

series data: stochastic processes

Economic and financial data can take one of three forms:

a) Cross-sectional data. At a given moment of time we observe oneor several variables corresponding to different economic orfinancial units. Usually microeconomic data.

b) Time series data. We observe one or several variables over time.These are often macroeconomic and financial variables.

c) Panel data. One or several variables corresponding to differenteconomic entities are observed over time.

In any case, the data can be univariate (only one variable is observed)or multivariate (several variables are observed).


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Cross-sectional observations are usually

assumed to be independent and identically

distributed (iid):

Independence: The order of the observations is

not important

Identically distributed: All observations are

generated by the same (univariate or multivariate)

random variable.


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Example of multivariate (bivariate) cross-sectional data:

i) Event study: Information content of quarterly earnings

announcements for firms in the Dow Jones Industrial Index

ii) Relation between expected returns and market betas.

2000

2400

Y vs. X

0

400

800

1200

1600

-400 400 8001200 2000 2800

X

Y


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A time series is a succession of observations of

one or several variables taken over time.

When just one variable is observed, we talk

about a univariate time series:Tyyy ,...,, 21

However, when several variables are observed

over time, we have a multivariate time series:

T

T

T

zzz

xxx

yyy

...

...

...

21

21

21


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Time series data are characterized by:a) Dependent. We can no longer consider that

the observation at time t is independent from

what we observe at time t-1. As aconsequence, and unlike what happens in thecross-sectional data, the order in which the

.b) The observations cannot be considered as

identically distributed: They are observed in a

context that evolves over time. We cannotconsider that we have T observations of thesame random variable .


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Examples of (univariate ) time series data

We may analyse historical earnings per share (EPS) to

forecast future EPS.

We may use a companys past returns to infer its risk:Prices and returns of SP500 observed daily from

4/1/1993 until 20/9/2011.

-.10

-.05

.00

.05

.10

.15

400

600

800

1,000

1,200

1,400

,

500 1000 1500 2000 2500 3000 3500 4000 4500

S&P500 RETURN_DAILY


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Example of multivariate time series: Monthly interest rates observed

from Sept. 1987 to Sept. 2006 for different maturities: i) 1-monthLondon interbank bid (LIBID) rate for US Dollars; ii) 3-month US

treasury-bill rates; iii) 6- 3-month US treasury-bill rates; iv) US

government bond rates for 1, 2, 3, 5, 7 and 10 years.


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In order to describe the mechanism generating atime series, we need to assume that we have arandom variable at each moment of time (astochastic process: succession of variables

ordered in time)

Then, each random variable generates oneobservation, obtaining our time series.


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Cross-sectional Time series

Univariate z1, z2,,zn

n independent observations from

the (unidimensional) variable Z

with (identical) distribution d(Z)

y1, y2,,yT

1 observation from the (univariate)

random process (Y(1), Y(2),,Y(T))

with joint distribution d(Y(1),

Y(2),,Y(T))

Multivariate z11, z12,,z1n

z21, z22,,z2n

y11, y12,,y1T

y21, y22,,y2T.

zk1, zk2,,zkn

n independent observations from

the (multidimensional) variable

(Z1, Z2,,Zk)

with (identical) (joint)distribution d(Z1, Z2,,Zk)

.

yk1, yk2,,ykT

1 observation from the

(multivariate) random process

(Y1(1), Y2(1),, YK(1),Y1(2),

Y2(2),Yk(2),, Y1(T), Y2(2),Yk(T))with joint distribution d(Y(1),

Y(2),,Y(T))


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1.3 Covariance stationarity and strict stationarity

A stochastic process can, in

principle, generate aninfinite number of

...

...

...

)(...)2()1(

)2()2(2

)2(1

)1()1(2

)1(1

T

T

yyy

yyy

TYYY

period t=1,,T............

...)3()3(

2)3(

1 Tyyy


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If we had several realizations of the process, we

could compute the mean of each randomvariable that constitutes the process according

m

ym

j

jt

t

==1

)(


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Unfortunately, in practice,

we only have a singlerealization, and Tyyy

TYYY

...

...

)(...)2()1(

21

compute the moments

of the random variables

that constitute the

process.


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In order to estimate the moments (mean,

variance etc) of every variable in time it isnecessary to restrict the properties of the

.

The restrictions that are usually imposed are

called stationarity.


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A process is (covariance, weakly) stationary if

i)

ii)

ttYE = ,))((

ttYVar


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The first condition allows us to estimate the mean (which is

common to all variables) using the sample mean:

yT

i

The same can be said about the other conditions.

Tyi

===1


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In practice, the conditions of stationarity allowus to estimate the autocorrelations (the mean

of the linear dependence between the

observations that are h periods apart) with

the sample autocorrelations

T

yyyy

hch

T

hthtt

== +=

1

))((

)()(


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Alternatively, we can define a strictly stationaryprocess as follows:

A stochastic process {Yt} is strictly stationary if themultivariate distribution function of {Yi,Yi+1,,Yi+k-1}

and {Xj,Xj+1,,Xj+k-1} are identical, for all integers i,j andfor all k>0.

A special example of a strictly stationary process is

Normal variables. All multivariate distributions arethen determined by the mean and varianceparameters.

However, a sequence of independent Normalvariables whose variance depend on the day of theweek is not a strictly stationary process.


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Weak stationarity does not imply strict

stationarity as only there is guarantee that thefirst two order moments are constant overtime.

Under Normality, weak stationarity implies.

Strict stationarity does not imply weak

stationarity unless the second order momentis finite.


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1.4 Correlation and independence: differences

between martingale difference, white noise, strictwhite noise and Gaussian white noise

The most popular measure of the dependence

between two random variables is thecorrelation which is given by

( )( )[ ]YX YXE

The correlation only measures lineardependence between the variables.

In the context of (stationary) time series, theautocorrelation is given by

YX

,

( )( )[ ] ( )( )2

][),(

=

=

htt

htt

hthttthtt

yyEyyEyyCorr


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In general, if the autocorrelations are zero we can only

conclude that there are is not a linear dependencebetween sucessive observations, i.e. we cannot predictthe expected future observations by looking at the pastevolution of the series.

However, zero autocorrelations do not implyindependence. It is possible that there are nonlinear

.

In a Gaussian process, if there is dependence betweensuccesive variables, this dependence can only be linear.Therefore, if the autocorrelations are zero (i.e. there isnot linear dependence), we can conclude that thevariables in the process are independent.


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Consider first an example of a nonlinear relation

between two random variables in a cross-sectionalexample

4

6

8

10

12y=x

2+a

The sample correlation between x and y is 0.1521 so,

there is not linear dependence. However, it is obious

that the variables are not independent.

-4 -3 -2 -1 0 1 2 3 4-4

-2

0

2


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In this case, x is a NID(0,1) variable but y is

clearly non-Normal

6

8

10

12

-3 -2 -1 0 1 2 3

-2

0

2

4

x

y


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Consider, now a time series example: the SP500 returns

described above. The marginal distribution (samplemoments) is given by

800

1,200

1,600

2,000

Series: RETURN_DAILYSample 1 4716Observations 4715

Mean 0.000215Median 0.000594Maximum 0.109572Minimum -0.094695Std. Dev. 0.012214Skewness -0.244155

The marginal distribution of daily returns is clearlyleptokurtic. Therefore, the marginal distribution is non-Normal and returns are not Gaussian.

0

400

-0.10 -0.05 0.00 0.05 0.10

.

Jarque-Bera 14503.73Probability 0.000000


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The sample autocorrelations of returns and of

squared returns are given by


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Therefore, daily returns are serially uncorrelated

but they are not independent. Their squares(volatilities) are correlated.

By looking to past evolution of daily returns, we

can predict the future evolution of daily

volatilities.


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Three categories of uncorrelated processes are of

particular importance in financial econometrics:White noise, strict white noise and martingaledifferences. These all are zero mean processes.

A process is white noise if i) it is stationary, ii)uncorrelated, and iii) has zero mean.

The absence of correlation from a white noisedoes not im l inde endence. The stron er

assumption that the variables are independentand identically distributed (iid) with zero means,defines strict white noise.

A martingale difference process has the followingfair playproperty

E[Yt|Y1,,Yt-1]=0


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Gaussian WN is SWN because uncorrelated

variables are independent when theirmultivariate joint distribution is Normal.

The distinction between WN and SWN isimportant when considering non-Gaussianprocesses as those required to model returns.

,

returns might have zero mean, be stationary ,be non-autocorrelated, (WN) and possesvolatility clustering. Then, the process is not

SWN because information about recentvolatility influences the variance ossubsequent returns.


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A MD is a zero mean, uncorrelated process.

Therefore, if it is further stationary, then it willbe WN.

WN may not be MD because the conditionalexpectation of an uncorrelated process can be

a nonlinear function

Bilinear process of Granger and Newbold (1986)which is a WN when 0


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1.5 Describing variables: Unconditional and conditional

distributionsThe complete description of the stochastic process is

given by thejoint distribution

When the joint distribution is multivariate Normal, the

rocess is said to be a Gaussian rocess.

( )Tyyd ,...,1

A Gaussian process is described by the mean andcovariance matrix given by

=

TTy

yy

E

2

1

2

1

[ ]

=

221

2

2

212

11221

21

2

1

TTT

T

T

T

T

yyy

y

y

y

E


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In practice, the dimension T is often very large

(in financial time series often over 2000).

Consequently, we look for alternative ways of

describing the distribution of the stochasticprocess.

There are two different univariate distributionsof interest: marginal and conditional

distributions.


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The marginal distribution is the scalar distribution of each of the variables in theprocess.

When the joint distribution is Normal (the process is Gaussian), the marginaldistribution of aech of the variables in the process is Normal. However, the Normality ofthe marginal distributions does not guarantee the joint Normality.

When dealing with univariate marginal distributions we lose information about thedependence.

The stationarity conditions refer to the moments of the marginal distribution.


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However, we know that there is dependence

between succesive observations. Therefore, itcould be of interest to analyse the distribution

of yt

conditional on y1

,,yt-1

.

If the rocess is Gaussian the conditional

distribution is Normal for all t. However, theconditional distribution can be Normal and

neither the marginal nor the joint being

Normal.


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Example 1

i) Consider the following model for a time series:

where is a strict white noise sequence with variance

In this case, the conditional mean is given by

ttt yy ++= 18.05

11

11 8.05][],...,|[

+== ttt

tt yyEyyyE

t2

The marginal mean is constant (stationarity condition)while the conditional mean evolves over time.

)8.01(5][8.05][ 1

=+= tt yEyE


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Consider now the conditional variance

The marginal variance is given by

( ) 221

2

1111 ))((),...,|( ===

t

tt

tt

ttt EyEyEyyyVar

2

2 ==

EVar

Note that the marginal variance is larger than

the conditional: using the information in the

past, we reduce the uncertainty about thefuture.

8.01


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If we further assume that is Gaussian, then

the conditional distribution of is also

Normal. Furthermore, given that is linear,

the marginal distribution is also Normal.

t

ty

ty


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Example 2

Consider the following process

Conditional mean

2

14.06.0 += ttt yy

0)(4.06.0)(2

1 =+=

ttt EyyE

Marginal mean

Both the conditional and the marginal mean areconstant over time and equal to zero.

0))(()(1

==

tt

t yEEyE


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Conditional variance

Marginal variance

2

1

2

1

2

1

2

114.06.0)()4.06.0()()(

+=+== tt

ttt

tt

tyEyyEyVar

]4.06.0[)]()4.06.0[()()(2

1

2

1

2

1

2 =+=+==

ttt

ttt yEEyEyEyVar

The marginal variance is constant (stationarity

condition) but the conditional varianceevolves over time.

1)4.01/(6.0=


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If we further assume that is Gaussian, thenthe conditional distribution of is also

Normal. However, given that is non-linear,

the marginal distribution is not Normal.

t

ty

ty


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1.6 Linear and non-linear models

Consider again the relationship between one variable andits own past evolution

yt=f(y1,,yt-1)+atwhere at is a white noise.

between yt and its own past evolution. If at is a strictwhite noise (possibily non-Gaussian), then there is notany further dependencies between yt and its past. Inthis case, we say that the model is lineal.

However, if at is an uncorrelated white noise with a non-Gaussian distribution, then it is possible that yt mayalso have non-linear dependencies with its past.


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In a linear model, if the conditional

distribution is Normal, the marginal is Normal. However, in nonlinear models, the conditional

can be Normal and the marginal being non-

Normal. Maravall (1983) shows that in a linear

,

autocorrelations of sqaures are equal to thesquared autocorrelations.

Note that a linear process can be also written

as a linear combination of past realizations ofthe innovations, .

t

chapter 1 financial statistics

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